Impact of chemical reaction on the Cattaneo–Christov heat flux model for viscoelastic flow over an exponentially stretching sheet

In this article, the numerical solutions for the heat transfer flow of an upper-convected Maxwell fluid across an exponentially stretched sheet with a chemical reaction on the Cattaneo–Christov heat flux model have been investigated. Using similarity transformation, the controlling system of nonlinear partial differential equations was transformed into a system of ordinary differential equations. The resulting converted equations were solved numerically by a successive linearization method with the help of MATLAB software. A graphic representation was created to analyze the physical insights of the relevant flow characteristics. The findings were presented in the form of velocity, temperature, and concentration profiles. As the relaxation time parameter varied, the local Nusselt number increased. The thermal relaxation time was shown to have an inverse relationship with fluid temperature. Furthermore, the concentration boundary layer becomes thinner as the levels of the reaction rate parameter increase. The results of this model can be applicable in biological fluids and industrial situations. Excellent agreement exists between the analysis's findings and those of the previous studies.


List of symbols x, y
Cartesian coordinates [m] (u, v) Velocity components  37 ), which is considered.
The formulation of the present problem is modelled with respect to following presumptions: 1. Upper-convected Maxwell fluid (UCM) flow 2. Micropolar liquid model 3. Cattaneo-Christov Heat Flux Model 4. Thermophoresis and chemical reaction effects are considered Under the above assumptions, the governing equations so obtained are given by Khan et al. 6 where V is the velocity vector, T is the Cauchy stress tensor and a is acceleration vector given by: The Cauchy stress tensor for a Maxwell fluid is: where the extra stress tensor S satisfies in which µ is the viscosity, 1 is the relaxation time, L is the velocity gradient, and the Rivlin-Ericksen tensor a 1 is defined through For a two-dimensional flow having velocity V one gets in the absence of pressure gradient the following equations in component form  www.nature.com/scientificreports/Using the boundary layer approximations 38 where δ being the boundary layer thickness, the flow is governed by Eq. ( 9) and where u and v represent the velocity's x− and y− directional components, respectively.The following relationship 39 holds when υ is the kinematic viscosity, 1 is the fluid relaxation time, T is the local fluid tempera- ture, k 1 (x) is the chemical reaction rate, and q is the heat flux which satisfies the following relationship where V is the velocity vector, k is the thermal conductivity, and 2 is the heat-flow relaxation time.We arrive at the following equations after eliminating q from Eqs. ( 3) and (4) (see  where α(= k/ρc P ) is the thermal diffusivity.

boundary conditions on velocity
The boundary sheet is assumed to be stretched with a large force in such a way that stretching velocity along the axial direction x is of exponential order of the directional coordinate.Hence, we employ the following boundary conditions on velocity (see Khan et al. 6 ).
Using the similarity transformations shown below 36 We see that similarity exists by substituting Eq. ( 17) into Eqs.( 11) -( 16), and we obtain the following: ∂u ∂x where Pr = υ α is the Prandtl number, S c = υ D is the Schmidt number, γ is the reaction rate parameter, and L is the non-dimensional fluid relaxation time and thermal relaxation time.The case of a Newtonian fluid is achieved when 1 = 0 in Eqs. ( 18)- (21).In addition, 2 = 0 fits the original Fourier law of heat conduction.
The skin friction coefficient C f defined as: The heat and mass transfers from the plate, respectively, are given by

Numerical methods
We employed SLM to solve the current problem numerically using MATLAB script file code.The SLM works by iteratively converting the controlling nonlinear Eqs. ( 18) -( 20) into a set of linear differential equations, which are then solved either analytically or numerically.
The SLM technique presupposes that the solutions of systems ( 18)-( 20) can be represented as 31,42 Starting from an initial guess that is appropriate for f 0 (η), θ 0 (η) and φ 0 (η) and satisfies boundary conditions (21), suitable functions are as follows: Substituting Eq. ( 22) into controlling Eqs. ( 18) - (20) while neglecting the nonlinear factors in f i (η), θ i (η) and φ i (η) and their derivatives yields depending on the conditions at the boundary, where Using the Chebyshev collocation spectral method 43 , the linearized system was solved, resulting in the system of equations below: We can write system (27) as matrix equation as www.nature.com/scientificreports/ The resultant system ( 28) is readily solved as

Results and discussion
This paper analyzed the effects of chemical reaction on the Cattaneo-Christov heat flux model for viscoelastic flow over an exponentially stretching sheet.Transfigured governing Eqs. ( 18) -( 20) with the boundary conditions (21) are coupled non-linear differential equations.Thus, it is impossible to solve directly with the analytical method.Therefore, to solve this coupled non-linear differential equations, we use SLM (SLM) method by MatLabR2023a software.For various values of effective governing parameters such as velocity ratio U, Deborah number 1 , Prandtl number Pr , Schmidt number S c , reaction rate parameter γ , and thermal relaxation time 2 , the numerical solutions of velocity, temperature, and concentration are obtained.The convergence of SLM solutions with respect to several orders of approximations for −f ′′ (0),−θ ′ (0) and −φ ′ (0) for different values of 1 when 2 = 0.5, A = 1.5,γ = 1, S c = 0.2, Pr = 1, is presented in Table 1.The comparison of the variation of the Nusselt number −θ ′ (0) for different values of 1 is presented in Table 2.The values show that our result is in admirable agreement with the results given by researchers Khan 6 in limiting conditions.Moreover, a comparison of different values of the Prandtl number Pr in the event that 1 = 2 = 0 , as well as the local Nusselt number θ ′ (0) for a range of parameter values are shown in Table 3.It can be observed that when the Prandtl number Pr and the parameter A are increased, the local Nusselt number θ ′ (0) also grows in magnitude.Furthermore, it has been discovered that there is a strong agreement between the current numerical values of the local Nusselt number θ ′ (0) and the numerical outcomes covered by Magyari and Keller 32 .Therefore, we are assured that for the analysis of our problem, the numerical method is appropriate.The SLM findings for the local Nusselt number, local Sherwood number, and skin friction coefficient are shown in Table 4 for various parameter values.
Figure 2 shows what happens to the hydrodynamic boundary layer when a fluid has a nondimensional relaxation time.An increase in 1 is interpreted as an increase in fluid viscosity.The fluid motion is resisted by increasing viscosity, which causes the velocity to decrease.Given that the Deborah number 1 is a good indicator of how long it will take a fluid to relax and come to rest when shear tension is eliminated, the thickness of the boundary layer likewise decreases for large 1 values.Many polymeric liquids that defy the Newtonian fluid model display these kinds of behaviors.The flow between two neighboring layers will decrease with an increase in Deborah number.Velocity and boundary layer thickness are generally reduced as a result.
Figure 3 illustrates the changes in Prandtl number Pr when considering the thermal relaxation time.With rising Pr , the thermal boundary layer's thickness and temperature decrease.which is qualitatively identical to the behavior of Pr on θ in both scenarios.In particular, the temperature variations of both the Fourier and Cat- taneo-Christov heat flux models have the same value as that of θ .Physically, the thermal diffusivity α and Prandtl number Pr are inversely correlated.The fluid is thought to experience less thermal influence as Pr increases.Therefore, when Pr increases, the thermal boundary layer becomes thinner.Owing to the thinner thermal bound- ary layer, the temperature profile is steeper, indicating that the wall slope of the temperature function is greater.
The effect of the temperature exponent A on the temperature profile is illustrated in Fig. 4.This figure shows the interesting 'Sparrow-Gregg hill' (SGH) phenomenon, in which temperature increases first reach their highest point before falling exponentially to zero.This implies that, for some negative reasons, reverse heat flow towards the sheet should be expected.The wall slope of the temperature function increased sharply as the positive/negative temperature exponent parameter A increased.
The effect of 1 on the thermal boundary layer is shown in Fig. 5.A larger 1 produces a stronger viscous force that resists the flow and raises the temperature.As a result, viscoelastic fluid has a higher temperature than a viscous fluid.
Figure 6 shows how the temperature distribution is affected by the nondimensional thermal relaxation time 2 .The thermal relaxation time and θ temperature have an inverse relationship.The temperature θ approached the free-stream condition at closer ranges above the sheet as 2 increased.In particular, both Newtonian and Maxwell fluids exhibit similar magnitudes of temperature θ changes with the thermal relaxation time.

Table 2.
Values of θ ′ (0) for different values of Pr, 1 , 2 and A compared to previous results Khan et al. 6 .www.nature.com/scientificreports/ Figure 7 shows that a decrease in concentration has been associated with an increase in Schmidt number S c .A lower mass diffusivity is associated with a lower Schmidt number S c .This elucidates why the thickness of the boundary layer concentration decreases as S c increases.
Figure 8 shows how a change in the reaction rate parameter γ affects the concentration profile.We observe that there is a noticeable decrease in concentration with an increase in γ .The contour in the free flow is uniformly

Conclusions
In this study, the impact of chemical reaction on the Cattaneo-Christov heat flux model for viscoelastic flow over an exponentially stretching sheet was investigated.governing system of nonlinear PDEs is transformed into a system of nonlinear ODEs using appropriate similarity transformations.The converted system equations were solved using SLM.The numerical results obtained agree very well with previously reported cases available in the literature.The following is a summary of the study's main findings: • In viscoelastic fluids, the hydrodynamic boundary layer is thinner than in viscous fluids.
• The thermal boundary layer thickness and temperature are decreasing functions of the relaxation time 2 .• For negative temperature exponent A, there are interesting Sparrow-Gregg Hills (SGH) for the temperature distribution.• Fourier's heat conduction law and the Cattaneo-Christov model's parameter responses are qualitatively comparable.• The concentration boundary layer becomes thinner as the levels of Schmidt number S c and reaction rate γ increase.• By setting 1 = 0 , we can retrieve the current consideration for the Newtonian fluid case.
• A few SLM iterations were enough to achieve great agreement with previous results.
Schmidt number t = 0 , unsteady fluid and mass flows begin.The sheet emerges from the origin through a slit and flows at the velocity of the U w (x) = U 0 e x L .The heating/cooling reference temperature T 0 is denoted by the variable surface temperature distribution T w = T ∞ + T 0 e Ax 2L (Magyari and Keller 36 ), and mass concentration C w = C ∞ + C 0 e x 2L ( Reddy et al. − SL T = µa 1 ,(6)a 1 = L + L T .

Figure 1 .
Figure 1.Physical model and coordinate system.
attenuated to a static value after the velocity climbs noticeably close to the wall.As a result, the chemical reaction speeds up the flow or increases the instantaneous transfer.The concentration boundary layer becomes thinner as the level of γ increase.

Figure 3 .
Figure 3.Effect of the Prandtl number Pr of on θ(η).

Table 3 .
Values of the θ ′ (0) for different values of Pr and A when 1 = 0 and 2 = 0 .The values in the brackets are from Magyari and Keller 36 .

Table 4 .
Using SLM, the effects of skin friction, the local Nusselt number, and the local Sherwood number were calculated for various parameter values.